Have we figured out if the OP wants to shoot targets or animals?
If targets then any reasonable caliber will do. My old man - who placed at Camp Perry shooting 600 yds with a service Springfield during the '30s - always said there are no inaccurate cartridges, just inaccurate rifles and, most important, inaccurate shooters.
500 yds is way too far to shoot game unless conditions are near perfect. No down range wind, no mirage, a standing broadside target and a good, solid rest. When I lived in the southwest many 500 yard shots presented themselves, but seldom (if ever) were conditions perfect. The longest game shot I've ever made was a measured 430 yds on a muley with a .243. It was early in the morning with no wind, resting over the hood of the pickup. I killed the deer but I never should have taken that shot. Game animals are too valuable and deserve our respect. There is a hell of a lot more to hunting than being able to shoot long distances. I'd rather develop the skills to get closer to the game and be sure of my shot.
Targets are a different thing of course. Shoot 'em from a mile away and have fun! For 500yds I'd choose a good 7mm mag with the best scope I could afford.
George